On Weakly Complete Group Algebras of Compact Groups

Karl Heinrich Hofmann and Linus Kramer∗

Communicated by K.-H. Neeb

Abstract. A topological vector space over the real or complex field K is weakly complete if it is isomorphic to a power KJ . For each topological group G there is a weakly complete topological group Hopf algebra K[G] over K = R or C, for which three insights are contributed: Firstly, there is a comprehensive structure theorem saying that the topological algebra K[G] is the cartesian product of its finite dimensional minimal ideals whose structure is clarified. Secondly, for a compact abelian group G and its character group Gb , the weakly complete complex b b b× b ∼ b b Hopf algebra C[G] is the product algebra CG with the comultiplication c: CG → CG G = CG ⊗CG , b Gb Gb c(F )(χ1, χ2) = F (χ1 + χ2) for F : G → C in C . The subgroup Γ(C ) of grouplike elements of b the group of units of the algebra CG is Hom(G,b (C \{0},.)) while the vector subspace of primitive ∼ b ∼ elements is Hom(G,b (C, +)). This forces the group Γ(R[G]) ⊆ Γ(C[G]) to be Hom(G,b S1) = Gb = G ∼ with the complex circle group S1 . While the relation Γ(R[G]) = G remains true for any compact ∼ group, Γ(C[G]) = G holds for a compact abelian group G if and only if it is profinite. Thirdly, for each pro-Lie algebra L a weakly complete universal enveloping Hopf algebra UK(L) over K exists such that for each connected compact group G the weakly complete real group Hopf algebra R[G] is a quotient Hopf algebra of UR(L(G)) with the (pro-)Lie algebra L(G) of G. The group Γ(UR(L(G))) of grouplike elements of the weakly complete enveloping algebra of L(G) ∼ maps onto Γ(R[G]) = G and is therefore nontrivial in contrast to the case of the discrete classical enveloping Hopf algebra of an abstract Lie algebra. Mathematics Subject Classification: 22E15, 22E65, 22E99. Key Words: Weakly complete vector space, weakly complete algebra, group algebra, Hopf algebra, compact group, Lie algebra, universal enveloping algebra.

1. Introduction For much of the material surrounding a theory of group Hopf algebras in the category of weakly complete real or complex vector spaces we refer to [3] and the forthcoming fourth edition of [6]. Some additional facts are presented here. A topological vector space over a locally compact field K is called weakly complete if it is isomorphic to KJ for some set J . This text considers K = R or K = C only and deals

∗Both authors were supported by Mathematisches Forschungsinstitut Oberwolfach in the program RiP (Research in Pairs). Linus Kramer is funded by the Deutsche Forschungsgemeinschaft under Germany’s Excellence Strategy EXC 2044-390685587, Mathematics Münster: Dynamics- Geometry-Structure.

ISSN 0949–5932 / $2.50 © Heldermann Verlag 408 Hofmann and Kramer with the categories G of topological groups, respectively, WA of weakly complete topological algebras. Inside each WA-object A we have the subset A−1 of all units (i.e., invertible elements) which turns out to be a G -object in the subspace topology. In [3] it was shown that the functor A → A−1 : WA → G has a left adjoint functor G 7→ K[G]: G → WA. Automatically, K[G] is a topological Hopf algebra. Then −1 for each topological group G there is a G -morphism ηG : G → K[G] such that for each WA-object A and G -morphism f : G → A−1 there is a unique WA-morphism ′ ′ f : K[G] → A such that f(g) = f (ηG(g)). The weakly complete algebra A = K[G] is seen to have a comultiplication c: A → A ⊗ A making it into a symmetric Hopf algebra. The subset G(A) = {a ∈ A−1 : c(a) = a ⊗ a} is a subgroup of A−1 and its elements are called grouplike. If G is a discrete group, then K[G] is the traditional group algebra over K and ηG is an embedding and its image is G(K[G]). In [3] it was shown that for a compact group G the map ηG is an embedding and for K = R induces an isomorphism onto G(R[G]). We shall see in this text that this fails over the complex ground field even for all compact abelian groups which are not profinite. The assignment G 7→ G(C[G]) may be viewed as as ‘complexification’ of the compact group G. This viewpoint becomes important if one considers the module category of C[G] in W , i.e. the representations of G on weakly complete complex vector spaces, but we will not pursue this in the present note. Since we shall deal with compact groups throughout this paper, we shall always consider G as a subgroup of the group K[G]−1 of units of K[G]. In particular, this means G ⊆ R[G] ⊆ C[G]. The article [3] and the 4th Edition of the book [6] identify a category H of weakly complete topological Hopf algebras for which the functor G → K[G] implements an equivalence of the category of compact groups and the category H. The vector space continuous dual of K[G] turns out to be the traditional representation algebra R(G, R). This approach yields a new access to the Tannaka-Hochschild duality of the categories of compact groups and “reduced” real Hopf algebras. In all of this, the precise nature of the weakly complete Hopf algebras K[G] even for compact groups remained somewhat obscure in the nondiscrete case. The present paper will present a precise piece of information on the topological algebra structure of K[G] in terms of a direct product of its finite dimensional minimal ideals whose precise structure links this presentation with the classical information of finite dimensional G-modules (cf. e.g. [6], Chapters 3 and 4). Certain complications have to be overcome on that level if one insists on an explicit identification of the algebra and ideal structure of R[G] as the case C[G] is easier. For abelian compact groups G we shall present a very direct access to the structure of the weakly complete Hopf algebra of C[G] by identifying an isomorphism between b C[G] and CG and by further identifying the group of grouplike elements to be isomorphic to (L(G), +) ⊕ G with the pro-Lie algebra L(G) of G (see [6]). The subgroup G of C[G]−1 therefore agrees with the group of grouplike elements of C[G] if and only if L(G) = {0} if and only if G is totally disconnected. The presence of weakly complete Lie algebras over K in the group K-Hopf algebra of a compact group motivates a proof of the existence of a weakly complete universal Hofmann and Kramer 409 enveloping algebra over K for WA Lie algebras over K. In contrast to the case of enveloping algebras on the purely algebraic side, the WA enveloping Hopf algebras will sometimes have grouplike elements. The universal property of the WA enveloping Hopf algebras will show that each WA Hopf-group algebra K[G] of a compact connected group G is a quotient algebra of the WA enveloping algebra of L(G). So WA enveloping algebras have a tendency of being larger than WA group algebras. For a special class of profinite dimensional Lie algebras a similar but different approach to appropriate enveloping algebras is considered in [4]. A preprint of the present material appeared in the Series of Preprints of the Mathe- matical Research Institute of Oberwolfach [5].

2. Weakly Complete Hopf Algebras For the basic theory of weakly complete Hopf algebras we may safely refer to [3] and [6], 4th Edition. For the present discussion we need a reminder of some basic concepts. Definition 2.1. Let A be a weakly complete symmetric Hopf algebra, i.e. a group object in the monoidal category (W, ⊗W ) of weakly complete vector spaces (see [6], Appendix 7 and Definition A3.62), with comultiplication c: A → A ⊗ A and coidentity k : A → K. An element a ∈ A is called grouplike if c(a) = a ⊗ a and k(a) = 1. The subgroup of grouplike elements in the group of units A−1 will be denoted G(A). An element a ∈ A is called primitive, if c(a) = a ⊗ 1 + 1 ⊗ a. The Lie algebra of primitive elements of ALie , i.e. the weakly complete Lie algebra obtained by endowing the weakly complete vector space underlying A with the Lie bracket obtained by [a, b] = ab − ba, will be denoted Π(A).

We recall that any weakly complete symmetric Hopf algebra A has an exponential → −1 function expA : A A as explained in [3], Theorem 3.12 or in [6], 4th edition, A7.41. Theorem 2.2. Let A be a weakly complete symmetric Hopf algebra. Then the following statements hold: (i) The set Γ(A) of grouplike elements of a weakly complete symmetric Hopf algebra A is a closed subgroup of (A, ·) and therefore is a pro-Lie group.

(ii) The set Π(A) of primitive elements of A is a closed Lie subalgebra of ALie and therefore is a pro-Lie algebra. ∼ G (iii) Π(A) = L( (A)) and the exponential function expA of A induces the exponen- → tial function expΓ(A) : Π(A) Γ(A) of the pro-Lie group Γ(A).

For a proof see e.g. [3], Theorem 6.15. Definition 2.3. For an arbitrary topological group G we define R(G, K) ⊆ C(G, K) to be that set of continuous functions f : G → K for which the linear span of the set of translations gf , gf(h) = f(hg), is a finite dimensional vector subspace of C(G, K). The functions in R(G, K) are called representative functions. 410 Hofmann and Kramer

Clearly R(G, K) is a subalgebra of C(G, K) also known as the representation algebra of G. In [3], Theorem 7.7(a) the following duality result was shown.

Theorem 2.4. (The Dual of a Weakly Complete Group Algebra K[G]) For an arbitrary topological group G, the function

′ FG : K[G] → R(G, K),FG(ω) = ω ◦ ηG is a natural isomorphism of Hopf algebras.

This applies, of course, to compact groups, in which case the Hopf algebra R(G, K) is a well-known object. For easy reference we record the following facts in the case of a compact group G for which we recall G ⊆ K[G]:

Theorem 2.5. For any compact topological group G, the following statements hold: (i) We have G ⊆ Γ(K(G)) ⊆ K[G]−1 (ii) In the case of K = R the equality G = Γ(R[G]) holds.

For (1) see [3], 5.4, and for (ii) see [3], 8.7. For the complex case, we shall see later in this paper that in many cases, a compact group G is a proper subgroup of Γ(C[G]).

3. Some preservation properties of K[−] Let us explicitly formulate and prove some preservation properties of our functor K[−]. Left adjoint functors preserve epics. A morphism of compact groups is an epimorphism if and only if it is surjective (see [6], RA3.17). Therefore the following lemma is to be expected.

Lemma 3.1. For every surjective morphism f : G → H of compact groups the morphism K[f]: K[G] → K[H] of weakly complete K- Hopf algebras is surjective. Proof. From the surjectivity of f : G → H we conclude that f(span(G)) = span(f(G)) = span(H) is dense in K[H] by Proposition 5.3 of [3], and likewise K(f)(K[G]) is dense in K[H]. But K[f] is, in particular, a W -morphism, that is, a morphism of weakly complete vector spaces. Every such has a closed image by [6], Theorem 7.30(iv). Hence K(f)(K[G]) = K[H]. This particular left adjoint functor K[−], however, also preserves the injectivity of morphisms:

Theorem 3.2. If G is a closed subgroup of the compact group H , then K[G] ⊆ K[H] (up to natural isomorphism). Proof. From the injectivity of a morphism of compact groups j : G → H we derive the surjectivity of C(j, K): C(H, K) → C(G, K) by the Tietze Extension Theorem. Now we set M := C(f, K) R(H, K)) ⊆ R(G, K). Since R(H, K) is dense in C(H, K) in the norm topology, M is dense in R(G, K) in the norm topology. Hofmann and Kramer 411

Then it is dense in L2(G, K) in the L2 -topology, and M is a G-module. In the case of K = R we can now apply Lemma 8.11 of [3] and conclude that M = R(G, R). Thus R(G, j): R(H, R) → R(G, R) is surjective. By Theorem 7.7 of [3] this implies that R[f]′ : R[H]′ → R[G]′ is surjective. The duality between K-vector spaces and weakly complete K-vector spaces shows that R[f]: R[G] → R[H] is injective. This proves the theorem for K = R. But then the commuting diagram

C⊗R[f] C ⊗R[G] −−−−−→ C ⊗R[H] ∼ ∼ =y y= C[G] −−−−−→ C[H] C[f] shows that C[f] is also injective. In the category of weakly complete vector spaces every injective morphism is an embedding by duality since every surjective morphism of vector spaces is a coretraction.

Corollary 3.3. Let G0 denote the identity component of the compact group G. Then

(i) The Hopf algebra K[G0] is a Hopf subalgebra of K[G]. ∼ (ii) R[G0] is algebraically and topologically generated by Π(R[G]) = L(G). Proof. (i) is a consequence of Theorem 3.2.

(ii) Note that the compact group G0 is algebraically and topologically generated R by expG(L(G)) (cf. [7], Corollary 4.22, p. 191), and that span(G0) = [G0] by [3], Corollary 5.3.

4. A principal structure theorem of K[G] Let G be a compact group and let E be a finite dimensional vector space over K ∈ {R, C}. We recall that the character of a representation ρ: G → EndK(E) is the continuous map g 7→ trK(ρ(g)). We also say that χ is the character of the G-module E . A representation, respectively, a G module, is determined by its character up to isomorphism. A character is called irreducible if the corresponding representation is irreducible (over the ground field K), equivalently, the corresponding G-module b is simple. We denote the set of all irreducible characters of G over K by GK . For each character χ of G, we select a finite dimensional G-module Eχ,K having χ as its character. If ε is an irreducible character, then the ring Lε,K = EndG(Eε,K) of all K-linear endomorphisms of Eε,K which commute with the G-action is, by Schur’s Lemma, a finite dimensional division ring over K. Hence

Lε,K = C if K = C,

Lε,K ∈ {R, C, H} if K = R, where H, as is usual, denotes the skew-field of quaternions. We view Eε,K as a right module over Lε,K . We denote the corresponding representation by → ⊆ ρε,K : G EndLε,K (Eε,K) EndK(Eε,K). Before we enter the presentation of the principal theorem on the weakly complete group algebra K[G] of a compact group we elaborate on some basic ideas of finite dimensional representation theory, indeed extending some of the presentation such 412 Hofmann and Kramer as it can be found e.g. in Chapter 3 of [6]. The first lemma extends the details of Proposition 3.21 of [6] and the comments which precede it.

Lemma 4.1. Let E be a finite dimensional vector space over K and ρ: G −→ EndK(E) an irreducible representation of a group G. Let A denote the K-span of the set {ρ(g) | g ∈ G}. Then A = EndL(E), where L = EndA(E) = EndG(E), L ∈ {R, C, H}.

Proof. First of all we note that A is a K-algebra containing idE . Hence every A- submodule of the additive group E is a G-invariant linear subspace, and vice versa. Therefore E is a simple A-module and so Jacobson’s Density Theorem applies, which, for the sake of completeness, we cite here in its entirety (see e.g. [2]).

Theorem 4.2. (Jacobson’s Density Theorem) Let M =6 0 be an (additive) abelian group, let A ⊆ End(M) be a subring and suppose that M is simple as a left A- module. Put L = EndA(M). Then L is a division ring and M is a right L-module 2k in a natural way. For every 2k-tuple (x1, . . . , xk, y1, . . . , yk) ∈ M , such that the elements x1, . . . , xk are linearly independent, there exists a ∈ A such that a(xi) = yi holds for all i = 1, . . . k.

Now the division ring L is a finite dimensional K-algebra over K, and hence is isomorphic to R, C, or H. Moreover, A ⊆ EndL(E). Let x1, . . . , xm be a L-basis for E , and let ϕ ∈ EndL(E) be arbitrary. Then there exists an element a ∈ A such that a(xi) = ϕ(xi) holds for all i = 1, . . . , m. Therefore ϕ = a and thus EndL(E) = A. The following result now extends [6], Lemma 3.14.

Lemma 4.3. Let E and F be finite dimensional vector spaces over K and suppose that ρ: G → EndK(E) and σ : H → EndK(F ) are irreducible representations of groups G, H . Suppose also that EndG(E) = L = EndH (F ). Then HomL(F,E) is an irreducible G × H -module over K, where (g, h)(f) = ρ(g) ◦ f ◦ σ(h−1).

Proof. We define A ⊆ EndK(E) and B ⊆ EndK(F ) as in Corollary 4.1. Then A = EndL(E) and B = EndL(F ). The K-vector space Hom(F,E) is in a natural way a right A-module and a left B -module. For every nonzero f ∈ HomL(F,E) we have AfB = HomL(F,E). Therefore HomL(F,E) is simple as G × H -module over K. We are now ready to prove a principal structure theorem for the weakly complete group algebra K[G] of a compact group G for either K = R or K = C. For each b ε ∈ GK we have the G-module Eε,K and the corresponding irreducible representation ρε,K : G → Endε,K(Eε,K) into the group of units of the concrete matrix ring Mε := L L L 2 Endε,K(Eε,K)) over = Qε,K of -dimension (dimL Eε,K) . Accordingly there is a → b unique function ρG : G ε∈GK Mε which is an injective group morphism into the multiplicative group of units of the product defined by the universal property of the b product such that the following diagram commutes for all χ ∈ GK : Q −−−−−→ρG G ∈b Mε  ε GK =y yprχ

G −−−−−→ Mχ ρχ,K Hofmann and Kramer 413

Theorem 4.4. For any compact group G the weakly complete symmetric Hopf algebra K[G] is a direct product Y K[G] = Kϵ[G] b ϵ∈GK b of finite dimensional minimal two-sided ideals Kϵ[G] such that for each ε ∈ GK there is a K-algebra isomorphism K ∼ ε[G] = Mε = EndLε,K (Eε,K).

In particular, each of these two-sided ideals Kε[G] is a two sided simple G×G-module and as an algebra is isomorphic to a full matrix ring over L.

Remark 4.5. The diagram

ηG G −−−−−→ K[G]   =y y∼ Q = −−−−−→ b G ε∈GK Mε  ρG  =y yprχ

G −−−−−→ Mχ ρχ,K b commutes for all χ ∈ GK . ′ ∼ Proof. By Theorem 2.4 and [6] Theorem 3.28, the topological dual K[G] = R(G, K) is the direct sum of the finite dimensional two-sided G-submodules Rϵ(G, K) b as ϵ ranges through the set of irreducible characters in GK . The G × G-module Rϵ(G, K) is defined in [6] as the image of the linear map ′ ⊗ −→ K ϕ : Eε,K K Eε,K R(G, ), where ϕ(u ⊗ v)(g) = hu, ρε,K(g)vi.

If we put ψ(f)(g) = trK(ρε,K(g)f), for f ∈ EndK(Eε,K) and g ∈ G, then the diagram ′ ⊗ −−−−−→ϕ K Eε,K K Eε,K Rε(G, )   sy =y ψ EndK(Eε,K) −−−−−→ Rε(G, K) commutes, where s(u ⊗ v) = [w 7→ vhu, wi]. We recall that group G × G acts on −1 Rε(G, K) via (a, b)(λ) = [g 7→ λ(a gb)]. If we put −1 −1 (a, b)(u ⊗ v) = (u ◦ ρε,K(a )) ⊗ ρε,K(b)v and (a, b)(f) = ρε,K(b) ◦ f ◦ ρε,K(a ), then all maps in this diagram are G × G-equivariant.

Suppose that K = L. Then EndK(Eε,K) = EndL(Eε,K) is simple as a G × G-module by Lemma 4.3 and thus ψ is an isomorphism. Suppose next that K(L. Then K = R and L = C or L = H. By the averaging process in [6] Lemma 2.15 there exists a G-invariant positive definite L-hermitian form (·|·) on E , semilinear in the first argument and linear in the second argument. 414 Hofmann and Kramer

This allows us to rewrite Rε(G, K) as the span of the maps g 7→ Re(u|gv), for u, v ∈ E . The G-invariance of (·|·) yields that Re(au|gbv) = Re(u|a−1gbv). If we consider the algebra inclusion

j : EndL(Eε,K) → EndK(Eε,K), then Re(u|v) = trK[w 7→ v(u|w)] holds for the trace map of EndK(Eε,K). It follows that the map ψ ◦ j in the diagram ′ ⊗ −−−−−→ϕ K Eε,K K Eε,K Rε(G, )   sy =y ψ Endε,Kx(Eε,K) −−−−−→ Rε(G, K)  j

EndL(Eε,K) is surjective and G × G-equivariant. Since EndL(Eε,K) is a simple G × G-module over K by Lemma 4.3, the map ψ ◦ j is an isomorphism. For the remaining part of the proof we apply standard duality theory. We put M χ R = Rε(G, K) b χ≠ ε∈GK

χ and define Kχ[G] as the annihilator of R . The annihilator mechanism supplies us with the diagram ⊥ K[ G] ←→ {0 }

⊥ K ←→ χ χ [G] R o ∼ = Rχ(G, K) ⊥ {0} ←→ R(G, K). ∼ Q By the duality of VK and WK it follows that K[G] = b K [G] with ϵ∈GK ϵ ∼ ′ Kϵ[G] = Rϵ(G, K) .

Now, if any closed vector subspace J of K[G] satisfies G·J ⊆ J and J ·G ⊆ J , then we also have span(G) · J ⊆ J and J · span(G) ⊆ J (where we view G as a subset of K[G]). Then Proposition 5.3 of [3] says that span(G) = K[G], and so K[G] · J ⊆ J and J · K[G] ⊆ J . That is, J is a closed two-sided ideal of K[G]. Therefore each Kϵ[G] is a two-sided ideal in K[G].

It remains to clarify the multiplicative structure of the ideals Kε[G]. If we consider b ε ∈ GK and the representation ρε,K , then the map −−−−−→ρε,K −1−−−−−→inc G GlLε,K (Eε,K) = EndLε,K (Eε,K) EndLε,K (Eε,K) and the universal property of K[G] described in the Weakly Complete Group Algebra Theorem 5.1 of [3] provides a morphism of weakly complete algebras K → πε : [G] EndLε,K (Eε,K) Hofmann and Kramer 415 extending ρε,K . We also have the product projection of weakly completeQ algebras K → K ′ b ′ prϵ : [G] ϵ[G]. Both maps πε and prϵ have the same kernel ϵ≠ ϵ ∈GK Aϵ . So there is an injective morphism K → α: ϵ[G] EndLε,K (Eε,K) ◦ such that πε = α prϵ . Since both algebras have the same dimension, α is an isomorphism of K-algebras.

Corollary 4.6. There is an isomorphism of G × G-modules M K R(G, ) = EndLε,K (Eε,K). b ε∈GK

Thus the multiplicity m of Eε,K as a G-module in R(G, K) is

dimK Eε,K m = dimLε,K (Eε,K) = . dimK L

This conclusion is well-known for K = C (see e.g Theorems 3.22 and 3.28 in [6]) but we could not readily find a reference for K = R. While the algebra structure of the weakly complete symmetric Hopf algebra K[G] is satisfactorily elucidated in Theorem 4.4, the comultiplication seems to be not easily accessible due to complications of the way how the representation theory of G × G reduces to that of G in general. In the case of commutative compact groups G and the complex ground field C these complications go away, and so we shall clarify the situation in these circumstances in the subsequent section.

5. The weakly complete group algebras of compact Abelian groups: an alternative view We have seen the usefulness of the concept of a weakly complete group algebra K[G] over the real or complex numbers. We obtained its existence from the Adjoint Functor Existence Theorem. This is rather remote from a concrete construction. It may therefore be helpful to see the whole apparatus in a much more concrete way at least for a substantial subcategory of the category of compact groups, namely, the category of compact abelian groups for which we already have a familiar duality theory due to Pontryagin and Van Kampen (see e.g. [6], Chapter 7). In this section let G be a compact abelian group and Gb = CAB(G, T) (with the category CAB of compact abelian groups and T = R/Z) its discrete character group. b b These groups are written additively. For K = C there is a natural bijection G → GC from the character group to the set of equivalence classes of complex simple G- modules (cf. [6], Lemma 2.30 (p.43), Exercise E3.10 (p.66)), and also Proposition b 3.56 (p.87) for some information on GR ). This bijection associates with a character ∈ b T C · 2πiχ χ G = Hom(G, ) the class of the moduleP Eχ = , χ c = e c. Accordingly, C C · ∈ b [6] Theorem 3.28 (12) reads R(G, ) = χ∈Gb fχ , for a suitable basis fχ , χ G, 2πiχ(g) ∼ (Gb) fχ(g) = e . In other words, as a G-module, R(G, C) = C . Accordingly, we expect C[G] to be uncomplicated. Our Theorem 4.4 makes this clear: 416 Hofmann and Kramer

The complex algebra C[G] may be naturally identified with the compo- b nentwise algebra CG .

b In the abelian case, our understanding of the comultiplication of C[G] = CG is much more explicit than in the general situation of Theorem 4.4. Each character χ: G → T −1 × 2πiχ(g) Gb determines a morphism fχ : G → C = C , fχ(g) = e , g ∈ G ⊆ C . By b the universal property of C[G] = CG , this value agrees with the χ-th projection of b g ∈ G ⊆ CG . Hence

b 2πi⟨χ,g⟩ (∀g ∈ G, χ ∈ G) ηG(g)(χ) = e . ∼ b ∼ Accordingly, if we write S1 = {z ∈ C; |z| = 1}, then g ∈ Hom(G,b S1) = Gb = G. Then in view of G ⊆ R[G] ⊆ C[G] we have

b b 1 b 1 G Hom(G, S ) ⊆ spanR(Hom(G, S )) = R[G] ⊆ C[G] = C . Recall from [3], Theorem 5.5 that we have an isomorphism

αG : C[G × G] → C[G] ⊗W C[G], and from [3] Lemma 5.12 we recall the comultiplication γG : C[G] → C[G] ⊗W C[G] to be the composition

δG αG C[G]−−−−−→C[G × G]−−−−−→C[G] ⊗W C[G].

Now for a compact abelian group G, the diagonal morphism δG : G → G × G has the group operation of Gb as its dual, namely: c b b b c δG : G × G → G. δG(χ1, χ2) = χ1 + χ2, as we write abelian group operations additively in general. If now we also write Gb×Gb C[G] ⊗W C[G] = C (identifying ϕ ⊗ ψ with (χ1, χ2) 7→ ϕ(χ1)ψ(χ2)), then we have c b b b b δG G G×G G γG = C : C → C , i.e., (∀ϕ ∈ C ), γG(ϕ)(χ1, χ2) = ϕ(χ1 + χ2).

b This allows us to determine explicitly the elements of the group G(CG) of all b b grouplike elements: Indeed a nonzero element ϕ ∈ CG is in G(CG) if and only if

Gb Gb Gb×Gb γG(ϕ) = ϕ ⊗ ϕ in C ⊗W C = C , where (ϕ ⊗ ϕ)(χ1, χ2) = ϕ(χ1)ϕ(χ2). This is the case if and only if b (∀ϕ1, ϕ2 ∈ G) ϕ(χ1 + χ2) = γG(ϕ)(χ1, χ2) = (ϕ ⊗ ϕ)(χ1, χ2) = ϕ(χ1)ϕ(χ2), that is, if and only if ϕ is a morphism of groups from Gb to C× = (C \{0}, ·). b Similarly, an element ϕ ∈ CG is primitive if and only if ϕ(χ1 + χ2) = γG(ϕ)(χ1, χ2) = (ϕ ⊗ 1) + 1 ⊗ ϕ) (χ1, χ2) = ϕ(χ1) + ϕ(χ2) if and only if ϕ: Gb → (C, +) is a morphism of topological groups. Let us summarize this discourse: Hofmann and Kramer 417

Theorem 5.1. (The Group Hopf Algebra C[G] for Compact Abelian G) Let G be a compact abelian group and A its weakly complete commutative symmetric group Hopf algebra C[G] and let Gb = Hom(G, T) be its character. b (i) Then A may be identified with CG such that g : Gb → A−1 is defined by (∀χ ∈ Gb) g(χ) = e2πi⟨χ,g⟩ ∈ S1,

where S1 = {z ∈ C : |z| = 1} ⊆ C× . The natural image of G in A−1 is

1 ∼ b 1 Gb G = Hom(G,b S ) = G,b and G = Hom(G,b S ) ⊆ R[G] ⊆ C[G] = C . (ii) If, as is possible in the category of weakly complete vector spaces, the weakly com- Gb×Gb plete vector spaces A ⊗W A and C are identified, then the comultiplication γG : A → A ⊗W A of A is given by b b (∀ϕ: G → C, χ1, χ2 ∈ G) γG(ϕ)(χ1, χ2) = ϕ(χ1 + χ2) ∈ C. (iii) The group of grouplike elements of A is b G(A) = Hom(G,b C×) ⊆ CG. (iv) The weakly complete Lie algebra of primitive elements of A is b P(A) = Hom(G,b C) ⊆ CG.

R× { ∈ C ∈ R ⊆ C} C× We write + for the multiplicative subgroup z : 0 < z of .

Corollary 5.2. For a compact abelian group G and the weakly complete commutative unital algebra A := C[G] we have a commutative diagram ∼ b b = P(A) = Hom(G, R) + Hom(G, iR) −−−−−→ L(G) × L(G)   × expAy yidL(G) expG G b R× · b S1 −−−−−→ × (A) = Hom(G, +) Hom(G, ) ∼ L(G) G. = The unique maximal compact subgroup of G(A) is G = Hom(G,b S1).

Proof. There is an elementary isomorphism of topological groups

(r, t + Z) 7→ ere2πit = er+2πit : R × T → C×.

× ∼ Accordingly, G(A) = Hom(G,b C ) = Hom(G,b R) ⊕ Hom(G,b T). ∼ Now Hom(G,b R) = Hom(R,G) (cf. [6], Proposition 7.11(iii)), Hom(R,G)=L(G) by b ∼ [6], Definition 5.7 (cf. Proposition 7.36ff., Theorem 7.66) and Hom(G,b T)=Gb = G by [6], Theorem 2.32. For the exponential function expA of a weakly complete unital symmetric Hopf algebra is treated in Theorem 2.2 above.

A compact abelian group is totally disconnected (i.e. profinite) if and only if L(G) = {0} (cf. [6], Corollary 7.72).

Remark 5.3. For a compact abelian group G the equality G = G(C[G]) holds if and only if G is totally disconnected (i.e. profinite). 418 Hofmann and Kramer

Proof. By Theorem 5.1, the equality holds if and only if L(G) = {0} if and only if Hom(G,b R) = {0} if and only if Gb is a torsion group (cf. e.g. [6], Propositions A1.33, A1.39) if and only if G is totally disconnected (see Corollary 8.5). In particular, e.g., T =6 G(C[T]). b Now we understand C[G] = CG rather explicitly, but R[G] only rather implicitly. However, Theorem 4.4 applies with K = R in order to shed some light on its intrinsic structure. b We define the function σG : C[G] → C[G] as follows: For χ ∈ G we set χˇ(g) = χ(−g) = −χ(g). Then we define b (∀ϕ ∈ CG) σ(ϕ)(χ) = ϕ(ˇχ).

Exercise 5.4. For a compact abelian group G, the function σG is an involution of weakly complete real algebras of C[G] whose precise fixed point algebra is R[G]. Accordingly, C[G] = R[G] ⊕ iR[G].

Remark 5.5. If ℵ is any cardinal and Gb is any abelian group with torsion free ∼ ℵ rank ℵ, then Hom(G,b R) = R . Proof. In [6], Theorem 8.20, pp.387ff. it is discussed that G contains totally disconnected compact subgroups ∆ such that the annihilator in the character group of G, say, ∆⊥ ⊆ Gb is free, and G/b ∆⊥ is a torsion group. This means that G/∆ is a torus. We note that the inclusion ∆⊥ → Gb induces an isomorphism ⊥ b b ⊥ ⊥ ∼ (X) K ⊗Z ∆ → K ⊗Z G and the (torsion free) rank of G is rank ∆ . If ∆ = Z for ⊥ ∼ a set X of cardinality rank ∆ , then Hom(G,b K) = KX .

b 5.1. The exponential function of C[G] = CG We recall from Theorem 2.2 that every weakly complete associative unital algebra b W has an exponential function, which is immediate in the case of W = CG as it is calculated componentwise. If the weakly complete algebra W is even a Hopf algebra, b such as CG , then the group Γ(W ) of grouplike elements is a pro-Lie group and Π(W ) is the (real!) Lie algebra of the pro-Lie group Γ(W ) ([3], Theorem 6.15). If indeed b W = CG = C[G] for a compact abelian group G, then the exponential function → expΓ(W ) : L(Γ(W )) Γ(W )) of Γ(W ) is the restriction of the (componentwise!) b b b exponential function exp: CG → (CG)−1 = ((C×,.))G to L(Γ(W )) = Hom(G,b C).

6. The weakly complete enveloping algebra of a weakly compact Lie algebra We have observed that for compact groups G the weakly complete real group algebra R[G] contains a substantial volume of different materials: the pro-Lie group G itself, its Lie algebra L(G), the exponential function between them and, as was discussed in detail in [3], a substantial portion of the Radon measure theory of G. The topological Hopf algebra K[G] is, in a sense, universally generated by G. So it seems natural to ask the question whether L(G) generates K[G] in a universal way – perhaps in some fashion that would resemble the universal enveloping algebra of a Lie algebra such Hofmann and Kramer 419 as it is presented in the famous Poincaré-Birkhoff-Witt-Theorem (see e.g. [1], §2, no 7, Théorème 1., p. 30). This is not exactly the case, but a few aspects should be discussed. So we let K again denote one of the topological fields R or C. Let WA denote the category of weakly complete associative unital algebras over K and and WL the category of weakly complete Lie algebras over K. The functor A 7→ ALie which associates with a weakly complete associative algebra A the weakly complete Lie algebra obtained by considering on the weakly complete vector space A the Lie algebra obtained with respect to the Lie bracket [x, y] = xy − yx is called the underlying Lie algebra functor. Since A is a strict projective limit of finite dimensional K -algebras by [3], Theorem 3.2, then ALie is a strict projective limit of finite dimensional K-Lie algebras, briefly called pro-Lie algebras. Every pro-Lie algebra is weakly complete. (Caution: A comment following Theorem 3.12 of [3] exhibits an example of a weakly complete K-Lie algebra which is not a pro-Lie algebra!)

Lemma 6.1. The underlying Lie algebra functor A 7→ ALie from WA to WL has a left adjoint U: WL → WA.

Proof. The category WL is complete. (Exercise. Cf. Theorem A3.48 of [6], p. 781.) The “Solution Set Condition” (of Definition A3.59 in [6], p. 786) holds. (Exercise: Cf. the proof of [3], Section 5.1 “The solution set condition”.) Hence U exists by the Adjoint Functor Existence Theorem (i.e., Theorem A3.60 of [6], p. 786).

In other words, for each weakly complete Lie algebra L there is a natural morphism λL : L → U(L) such that for each continuous Lie algebra morphism f : L → ALie for a weakly complete associative unital algebra A there is a unique WA-morphism ′ → ′ ◦ f : U(L) A such that f = fLie λL . WL WA

λL L −−−−−→ U(L)Lie U(L)    ∀ ′ ∃ ′ fy yfLie y !f

ALie −−−−−→ ALie A. id

If necessary we shall write UK instead of U whenever the ground field should be emphasized. We shall call UK(L) the weakly complete enveloping algebra of L (over K).

Example 6.2. Let L = K, the smallest possible nonzero Lie algebra over K. Then U(L) = KhXi (see [3], Definition following Corollary 3.3), and define λL : L → U(L)Lie by λL(t) = t·X . Indeed the universal property is satisfied by [3], Corollary 3.4. Namely, let f : K → ALie be a morphism of weakly complete Lie algebras. Then there is a unique morphism f ′ : U(L) → A such that f ′(X) = f(1) by [3], Corollary 3.4. Then f ′(t·X) = t·F ′(X) = t·f(1) = f(t). 420 Hofmann and Kramer

Thus by Lemma 3.5 of [3] and the subsequent remarks we have:

The weakly complete enveloping algebra UC(C) over C of the smallest nonzero complex Lie algebra is isomorphic to the weakly complete commutative algebra C[[X]]C with the complex power series algebra ∼ N0 C[[X]] = C , N0 = {0, 1, 2,... }. The size of the weakly complete enveloping algebras therefore is considerable.

Proposition 6.3. The universal enveloping functor U is multiplicative, that is, × → ⊗ there is a natural isomorphism αL1L2 : U(L1 L2) U(L1) W U(L2). Proof. We have a natural bilinear inclusion map of weakly complete vector spaces j : U(L1) × U(L2) → U(L1) ⊗W U(L2) yielding

λ1×λ2 j L1×L2−−−−−→ U(L1)Lie× U(L2)Lie−−−−−→ U(L1)Lie⊗W U(L2)Lie and U(L1)Lie ⊗W U(L2)Lie=(U(L1) ⊗W U(L2))Lie, the composition α0 of which is a morphism of weakly complete Lie algebras. Hence the universal property yields a morphism of weakly complete associative algebras

(1) α: U(L1 × L2) → U(L1) ⊗W U(L2) ◦ ⊗ such that α0 = αLie λL1 λL2 .

The functorial property of U allows us to argue that each of U(Lm), m = 1, 2 is a retract of U(L1 × L2) so that we may assume U(Lm) ⊆ U(L1 × L2), m = 1, 2. Now the multiplication in U(L1 × L2) gives rise to a continuous bilinear map U(L1)×U(L2) → U(L1 ×L2), and then the universal property of the tensor product of weakly complete vector spaces yields the morphism

(2) β : U(L1) ⊗W U(L2) → U(L1 × L2). Similarly to the proof of [3], Theorem 5.5 (preceding the statement of the theorem) we argue that α and β are inverses of each other, and so α of (1) is the desired isomorphism αL1L2 .

Lemma 6.4. For any weakly complete unital algebra A, the vector space morphism ∆A : A → A ⊗W A, ∆A(a) = a ⊗ 1 + 1 ⊗ a is a morphism of weakly complete Lie algebras ALie → (A ⊗W A)Lie . Proof. Since the functions a 7→ a⊗1, 1⊗a are morphisms of topological vector spaces, so is ∆A For y1, y2 ∈ A, write zj := ∆A(yj) = yj ⊗ 1 + 1 ⊗ yj Then just as in the classical case, we calculate

[∆A(y1), ∆A(y2)] = [z1, z2] = z1z2 − z2z1

= (y1 ⊗ 1 + 1 ⊗ y1)(y2 ⊗ 1 + 1 ⊗ y2)

− (y2 ⊗ 1 + 1 ⊗ y2)(y1 ⊗ 1 + 1 ⊗ y1)

= (y1y2 ⊗ 1 + y1 ⊗ y2 + y2 ⊗ y1 + 1 ⊗ y1y2)

− (y2y1 ⊗ 1 + y2 ⊗ y1 + y1 ⊗ y2 + 1 ⊗ y2y1)

= [y1, y2] ⊗ 1 + 1 ⊗ [y1, y2] = ∆A[y1, y2].

Thus ∆A is a morphism of Lie algebras as asserted. Hofmann and Kramer 421

Now we consider a weakly complete Lie algebra L and recall that λL : L → U(L)Lie is a morphism of weakly complete Lie algebras. Thus by Lemma 6.4,

pL = ∆U(L) ◦ λL : L → (U(L) ⊗ U(L))Lie is a morphism of weakly complete Lie algebras. Now by the universal property of U, pL induces a unique natural morphism of weakly complete associative unital algebras γL : U(L) → U(L) ⊗W U(L) such that pL = (γL)Lie ◦ λL . This yields the following insight, where kL : L → {0} denotes the constant morphism.

Corollary 6.5. Each weakly complete enveloping algebra U(L) is a weakly complete Hopf algebra with the comultiplication γL and the coidentity U(kL): U(L) → K.

Proof. Observe that γL is a morphism of weakly complete unital algebras satisfying γL(y) = y ⊗ 1 + 1 ⊗ y for y = λ(x), x ∈ L. The associativity of this comultiplication is readily checked as in the case of abstract enveloping algebras. The constant morphism of weakly complete Lie algebras L → {0} yields a morphism of weakly complete unital algebras U(L) → U({0}) = K which is the coidentity of the Hopf algebra.

Our results from [3] regarding weakly complete associative unital algebras and Hopf algebras over K apply to the present situation.

Theorem 6.6. (The Weakly Complete Enveloping Algebra) Let L be a weakly complete Lie algebra. Then the following statements hold: (i) U(L) is a strict projective limit of finite-dimensional associative unital algebras and the group of units U(L)−1 is dense in U(L). It is an almost connected pro-Lie group (which is connected in the case of K = C). The algebra U(L) −1 has an exponential function exp: U(L)Lie → U(L) ,

(ii) The pro-Lie algebra Π(U(L)) of primitive elements of U(L) contains λL(L),

(iii) The subalgebra generated by λL(L) in U(L) is dense in U(L). (iv) The pro-Lie algebra Π(U(L)) is the Lie algebra of the pro-Lie group Γ(U(L)) of grouplike elements of U(L). We use the abbreviation G := Γ(U(L)) and → note that the exponential function expG : L(G) G is the restriction and corestriction of the exponential function exp of U(L) to Π(U(L)), respectively, G. The image exp(L(G)) generates algebraically and topologically the identity component G0 of G. Proof. (i) See [3], Theorems 3.2, 3.11, 3.12, 4.1.

(ii) The very definition of the comultiplication γL for Corollary 6.5 shows that for any y ∈ λL(L), the image under the comultiplication γL is y ⊗ 1 + 1 ⊗ y, which means that y is primitive. (iii) An argument analogous to that in the proof of Proposition 5.3 of [3] showing that, for the case of any topological group T , the subset ηG(T ) of the weakly complete group algebra K[T ] spans a dense subalgebra, shows here that the closed subalgebra S generated in U(L) by λL(L) has the universal property of U(L) and therefore agrees with U(L). (iv) See Theorem 2.2 and [3], Theorem 6.15. 422 Hofmann and Kramer

Remark 6.7. We note right away that for any weakly complete Lie algebra L which has at least one nonzero finite dimensional K-linear representation, the morphism λL : L → U(L)Lie is nonzero. By Ado’s Theorem, this applies, in particular, to any Lie algebra which has a nontrivial finite dimensional quotient and therefore is true for the Lie algebra L(P ) of any pro-Lie group P .

Corollary 6.8. (i) The weakly complete enveloping algebra U(L) of a weakly complete Lie algebra L with a nontrivial finite dimensional quotient has nontrivial grouplike elements.

(ii) If L is a pro-Lie algebra, then λL : L → U(L)Lie maps L isomorphically onto a closed Lie subalgebra of the pro-Lie algebra Π(U(L)) of primitive elements.

Proof. (i) If Π(U(L)) is nonzero, then U(L) has nontrivial grouplike elements by Theorem 6.6(iii), and by (ii) of 6.6, this is the case if λL is nonzero which is the case for all L satisfying the hypothesis of the Corollary by the remark preceding it. (ii) Since each finite dimensional quotient of L has a faithful representation by the Theorem of Ado, and since the finite dimensional quotients separate the points of L, the morphism λL is injective. However, injective morphisms of weakly complete vector spaces are open onto their images.

It follows that for pro-Lie algebras L we may assume that L is in fact a closed Lie subalgebra of primitive elements of U(L) which generates U(L) algebraically and topologically as a weakly complete algebra. It remains an open question under which circumstances we then have in fact L = P(U(L)). In the classical setting of the discrete enveloping Hopf algebra in charac- teristic 0 this is the case: see e.g. [8], Theorem 5.4.

One application of the functor U is of present interest to us. Recall that for a compact group we naturally identify G with the group of grouplike elements of R[G] (cf. [3], Theorems 8.7, 8.9 and 8.12), and that L(G) may be identified with the pro-Lie algebra Π(R[G]) of primitive elements. (Cf. also Theorem 2.2 above.) We may also assume that L(G) is contained the set Π(U(L(G))) of primitive elements of UR(L(G)).

Theorem 6.9. (i) Let G be a compact group. Then there is a natural morphism of weakly complete algebras ωG : UR(L(G)) → R[G] fixing the elements of L(G) elementwise.

(ii) The image of ωG is the closed subalgebra R[G0] of R[G].

(iii) The pro-Lie group Γ(UR(L(G))) is mapped onto G0 = Γ(R[G0]) ⊆ R[G]. The connected pro-Lie group Γ(UR(L(G)))0 maps surjectively onto G0 and P(UR(L(G))) onto P(R[G]).

Proof. (i) follows at once from the universal property of U.

(ii) As a morphism of weakly complete Hopf algebras, ωG has a closed image which is generated as a weakly complete subalgebra by L(G) which is R[G0] by Corollary 3.3 (ii). Hofmann and Kramer 423

(iii) The morphism ωG of weakly complete Hopf algebras maps grouplike elements to grouplike elements, whence we have the commutative diagram

Π(ωG) L(G) ⊆ Π(UR(L(G))) −−−−−→ Π(R(G))= L(G)   expΓ(UR(L(G)))y yexpG

Γ(UR(L(G))) −−−−−→ Γ(R[G]) = G. Γ(ωG) P Since (ω) is a retraction and the image of expG topologically generates G0 , the G ◦ image of (ωG) expUR(L(G)) topologically generates G0 . Since the image of the exponential function of the pro-Lie group G(UR(L(G))) generates topologically its identity component, G(ωG) maps this identity component onto G0 .

Since L(G) ⊆ P(UR(L(G)), and since any morphism of Hopf algebras maps a primitive element onto a primitive element we know ωG(P(UR(L(G)))) = P(R[G]).

It remains an open question whether G(UR(L(G))) is in fact connected. An overview of the situation may be helpful: R[ G]

ωG,onto UR(L (G)) −−−−−→ R[G 0] |

Γ(UR (G)) −−−−−→ −−−−−→ Γ(R[G ]) = G

onto Γ(UR(xL(G)))0 −−−−−→ Γ(R[Gx0])=G0 = Gx0    exp exp  UR(L(G))  R[G] = expG

Π(UR(L(G))) = Π(R[G0])=Π(R[G]) = L(G).

A noteworthy consequence of the preceding results is the insight that Q RX × for any nonzero weakly complete real Lie algebra L = j∈J Lj for any set X and any family of compact finite dimensional simple Lie algebras Lj , the weakly complete enveloping algebra U(L) has grouplike elements. In the discrete situation, the enveloping algebra U(L) of a Lie algebra L for char- acteristic zero has no grouplike elements.

Acknowledgments. We thank the referee for carefully reading our manuscript and so eliminating a good number of typos and flaws. – An essential part of this text was written while the authors were partners in the program Research in Pairs at the Mathematisches Forschungsinstitut Oberwolfach MFO in the Black Forest from February 4 through 23, 2019. The authors are grateful for the environment and infrastructure of MFO which made this research possible. – The authors also acknowledge the joint work with Rafael Dahmen of the Karlsruhe Institut für Technologie; a good deal of that work did take place at the Technische Universität Darmstadt in 2018). – The second author thanks Siegfried Echterhoff for helpful insights. 424 Hofmann and Kramer

References [1] N. Bourbaki: Groupes et Algèbres de Lie, Hermann, Paris (1971). [2] C. W. Curtis, I. Reiner: Representation Theory of Finite Groups and Associative Al- gebras, Pure and Applied Mathematics XI, Interscience Publishers, Wiley & Sons, New York (1962). [3] R. Dahmen, K. H. Hofmann: The pro-Lie group aspect of weakly complete algebras and weakly complete group Hopf algebras, J. Lie Theory 29 (2019) 413–455. [4] A. Hamilton: A Poincaré-Birkhoff-Witt theorem for profinite pronilpotent Lie ,algebras J. Lie Theory 29 (2019) 611–618. [5] K. H. Hofmann, L. Kramer: Group algebras of compact groups, preprint of the Mathe- matische Forschungsinstitut Oberwolfach (2019) 30 pp. [6] K. H. Hofmann, S. A. Morris: The Structure Theory of Compact Groups – a Primer for the Student – a Handbook for the Expert, 3rd ed., De Gruyter Studies in Mathematics 25, Berlin (2013). [7] K. H. Hofmann, S. A. Morris: The Lie Theory of Connected Pro-Lie Groups, – a Struc- ture Theory for Pro-Lie Algebras, Pro-Lie Groups, and Connected Locally Compact Groups, European Mathematical Society Publishing House, Zürich (2006). [8] J.-P. Serre: Lie Algebras and Lie Groups, W. A. Benjamin, New York (1965).

Karl Heinrich Hofmann Linus Kramer Fachbereich Mathematik Mathematisches Institut Technische Universität Darmstadt Universität Münster 64289 Darmstadt, Germany 48149 Münster, Germany [email protected] [email protected]

Received August 15, 2019 and in final form November 11, 2019